'.'I PROCEEDINGS OF THE AMERICAN ACADEMY. 



where k Is any integer, and of Gj by the equations 



= „ + u 2 «j _ «j _ 



e "tT«» _ j L «. u 2 /J 



(17 a) 



Oj s= a., -\- a 2 . 



Nevertheless, for any value of k, the parameter group of G£ is identical 

 with the parameter proup of G% . For let S a denote the transformation 



(21 



defined by equations (13 a), and S~ the transformation defined by 

 equations ( 17 a). Then for any system of values of a l5 a 2 , aud for any 

 value of k, we have, by properly choosing [S x , (3 2 , 



C.O v' - 



S a = S fi '> 



which symbolic equation persists for /3i, /? 2 , and k arbitrary, if a%, a 2 are 

 properly chosen.* 



Let T a lie an arbitrary transformation of G r , defined by the equations 



1 <• r 



*i = *t + 2, a, A', X t + — 2, 2 4 a, a t A X 4 X,. + . . . 



i -i : l i 



(/= 1,2 . . . n). 



The transformation T a , inverse to T a , is then defined by the equa- 

 tions which we obtain on replacing a lf a 2 • . . by their negatives.! 

 Compound the transformation T a and its inverse with each transforma- 

 tion T a of G r so as to obtain the transformation T a T a T' a , which is also 

 a transformation of G r ; aud let 



?* = T a r„r a -\t 



( l .' ) /,,.. = / fl /„. / g . 



Then, if 



7; =y.7',. 



we have 



(2i) /• / /:,7' y "\ 



1 18 a) may l»- regarded u defining a group with three parameters 

 a| , a., and /■, nf which two, a, and a., vary continuously, and one, namely /, takes 

 only integer values. Bui this group is nol a mixed group, since we have shown 

 that / is unessential, thai is, it i- immaterial what value is assigned to k. 

 t TransformAtionsgruppen, I. 52 



1 The transformation '/'.,• is said by Lie to be obtained by the application ( I 

 of '/', to the transformation! / of G r . Q Lie Continuierliche Grup- 

 pen ■ I y. 



